## Math

Patterns & Sorting Kindergarten Math Patterns & Sorting Kindergarten Math Over & Under Kindergarten Math On & Off Kindergarten Math Shapes Kindergarten Math Whole Numbers First Grade Math Patterns & Sorting Kindergarten Math ### N.1. Number and Operations

#### 1.1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

##### 1.1.1. Work flexibly with fractions, decimals, and percents to solve problems.

##### 1.1.2. Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.

##### 1.1.3. Develop meaning for percents greater than 100 and less than 1.

##### 1.1.4. Understand and use ratios and proportions to represent quantitative relationships.

##### 1.1.5. Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.

##### 1.1.6. Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

##### 1.1.7. Develop meaning for integers and represent and compare quantities with them.

#### 1.2. Understand meanings of operations and how they relate to one another.

##### 1.2.1. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

##### 1.2.2. Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.

##### 1.2.3. Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

#### 1.3. Compute fluently and make reasonable estimates.

##### 1.3.1. Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.

##### 1.3.2. Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.

##### 1.3.3. Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

##### 1.3.4. Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

### N.11. Grade 7 Curriculum Focal Points

#### 11.1. Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity

##### 11.1.1. Students extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multi-step problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).

#### 11.2. Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes

##### 11.2.1. By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base x Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three-dimensional shapes to model real-world situations and solve a variety of problems (including multi-step problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.

#### 11.3. Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations

##### 11.3.1. Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.

### N.12. Connections to the Grade 7 Focal Points

#### 12.1. Measurement and Geometry: Students connect their work on proportionality with their work on area and volume by investigating similar objects. They understand that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related. Students apply their work on proportionality to measurement in different contexts, including converting among different units of measurement to solve problems involving rates such as motion at a constant speed. They also apply proportionality when they work with the circumference, radius, and diameter of a circle; when they find the area of a sector of a circle; and when they make scale drawings.

#### 12.2. Number and Operations: In grade 4, students used equivalent fractions to determine the decimal representations of fractions that they could represent with terminating decimals. Students now use division to express any fraction as a decimal, including fractions that they must represent with infinite decimals. They find this method useful when working with proportions, especially those involving percents. Students connect their work with dividing fractions to solving equations of the form ax = b, where a and b are fractions. Students continue to develop their understanding of multiplication and division and the structure of numbers by determining if a counting number greater than 1 is a prime, and if it is not, by factoring it into a product of primes.

#### 12.3. Data Analysis: Students use proportions to make estimates relating to a population on the basis of a sample. They apply percentages to make and interpret histograms and circle graphs.

#### 12.4. Probability: Students understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of outcomes in which the event occurs. Students use theoretical probability and proportions to make approximate predictions.

### N.2. Algebra

#### 2.1. Understand patterns, relations, and functions.

##### 2.1.1. Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

#### 2.2. Represent and analyze mathematical situations and structures using algebraic symbols.

##### 2.2.1. Develop an initial conceptual understanding of different uses of variables.

##### 2.2.2. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

##### 2.2.3. Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

##### 2.2.4. Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

#### 2.3. Use mathematical models to represent and understand quantitative relationships.

##### 2.3.1. Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

#### 2.4. Analyze change in various contexts.

##### 2.4.1. Use graphs to analyze the nature of changes in quantities in linear relationships.

### N.3. Geometry

#### 3.1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

##### 3.1.1. Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

##### 3.1.2. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

##### 3.1.3. Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

#### 3.2. Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

##### 3.2.1. Use coordinate geometry to represent and examine the properties of geometric shapes.

##### 3.2.2. Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.

#### 3.3. Apply transformations and use symmetry to analyze mathematical situations.

##### 3.3.1. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

##### 3.3.2. Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.

#### 3.4. Use visualization, spatial reasoning, and geometric modeling to solve problems.

##### 3.4.4. Use geometric models to represent and explain numerical and algebraic relationships.

### N.4. Measurement

#### 4.1. Understand measurable attributes of objects and the units, systems, and processes of measurement.

##### 4.1.1. Understand both metric and customary systems of measurement.

##### 4.1.2. Understand relationships among units and convert from one unit to another within the same system.

##### 4.1.3. Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

#### 4.2. Apply appropriate techniques, tools, and formulas to determine measurements.

##### 4.2.2. Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

##### 4.2.3. Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

##### 4.2.4. Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

##### 4.2.5. Solve problems involving scale factors, using ratio and proportion.

##### 4.2.6. Solve simple problems involving rates and derived measurements for such attributes as velocity and density.

### N.5. Data Analysis and Probability

#### 5.1. Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

##### 5.1.1. Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.

##### 5.1.2. Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

#### 5.2. Select and use appropriate statistical methods to analyze data.

##### 5.2.1. Find, use, and interpret measures of center and spread, including mean and interquartile range.

##### 5.2.2. Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

#### 5.3. Develop and evaluate inferences and predictions that are based on data.

##### 5.3.2. Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

##### 5.3.3. Use conjectures to formulate new questions and plan new studies to answer them.

#### 5.4. Understand and apply basic concepts of probability

##### 5.4.2. Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.

##### 5.4.3. Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

### N.6. Problem Solving

#### 6.1. Build new mathematical knowledge through problem solving.

#### 6.2. Solve problems that arise in mathematics and in other contexts.

#### 6.3. Apply and adapt a variety of appropriate strategies to solve problems.

### N.7. Reasoning and Proof

#### 7.1. Recognize reasoning and proof as fundamental aspects of mathematics.

#### 7.2. Make and investigate mathematical conjectures.

#### 7.3. Develop and evaluate mathematical arguments and proofs.

#### 7.4. Select and use various types of reasoning and methods of proof.

### N.9. Connections

#### 9.2. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

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